
\documentclass[12pt]{article}

\usepackage{amsmath}

\setlength{\textheight}{9.5truein}
\addtolength{\topmargin}{-1.0truein}

\setlength{\oddsidemargin}{0.1truein}
\setlength{\evensidemargin}{0.1truein}
\setlength{\textwidth}{6.3truein} 

\newcommand{\commentt}[1]{}

\newcommand{\vsph}{\vspace{.40cm}}

\newcommand{\vsp}{\vspace{.40cm}}
\newcommand{\vecc}[1]{\mbox{\boldmath $#1$}}

\newcounter{qnum} \newcounter{pnum}[qnum]




\newcommand{\QQ}{\vsph \hrule \vsph \stepcounter{qnum} {\noindent \bf
    Problem \theqnum.} \hspace{.2cm} }

\newcommand{\PP}{\stepcounter{pnum} \item[{\bf (\alph{pnum})}]\, }

\newenvironment{parts}{
  \begin{list}{}{\setlength{\topsep}{2.0ex}\setlength{\itemsep}{1.5ex}
      \setlength{\leftmargin}{2.3em}\setlength{\labelsep}{0.8em}}
  }{\end{list}}

\newcommand{\cov}{\mbox{Cov}} \newcommand{\var}{\mbox{Var}}

\newenvironment{clist} {\vspace{-.5cm}
  \begin{list}{$\bullet$} {\setlength{\topsep}{.5cm}
      \setlength{\itemsep}{.1cm} \setlength{\parsep}{0cm}}
  }{\end{list}}


\begin{document}


\begin{center}
  \bf \large
  M.\ S.\ Comprehensive Exam\\
  Thursday, August 21, 2008
\end{center}


\vsp

You have three hours.  Do as many problems as you can.  No one is
expected to answer all the problems correctly.  Partial credit will be
given.  All problems are worth an equal amount of credit.

\vsp

{\bf Put your solution to each problem on a separate sheet of paper.}


\vsph

%%%%%%%%%%%%%%%%%%%% Applied Statistics %%%%%%%%%%%%%%%%%%%%%%

% Adrian, regression.

\QQ Answer the following questions:
\begin{parts}

  \PP We have a linear regression problem with $n$ observations
\[
y_i=\beta_0+\beta_1x_{i1}+ \cdots +\beta_kx_{ik}+\epsilon_i, i=1,...,n
\]
which can be written in matrix form as $Y=X\beta+\epsilon$. Assuming
that $X,Y$ are given matrices of real numbers and $\epsilon_i$ are
i.i.d.\ from ${\cal N}(0,\sigma^2)$, find the least squares estimators
of $\beta$ and derive their distribution.

\PP What is a graphical tool for checking the normality of the
residuals?

\PP To find outliers, jackknife residuals (named studentized residuals
in SAS) are computed for each observation and compared to a
cutoff. What type of outliers can{\bf not} be detected using this
method?

\end{parts}


% Eric: Applied:

\QQ The following regression situation is observed:
$$
Y_{ij} = \beta_0 + \beta_1 X_{i} + \varepsilon_{ij}, \hspace{.1in} i=1, 2, \ldots, l, \hspace{.1in} j=1, 2, \ldots n_i.
$$
The errors are independent normal (0, $\sigma^2$) random variables. We wish to see if this model adequately fits the data by performing a lack of fit test. The collected data is shown in the table below.
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|l|}\hline
$X$ Level & Responses \\
\hline
$X_1$ & $Y_{11}, Y_{12}, Y_{13}$ \\
$X_2$ & $Y_{21}$ \\
$X_3$ & $Y_{31}, Y_{32}$ \\
$X_4$ & $Y_{41}, Y_{42}, Y_{43}$ \\
$X_5$ & $Y_{51}, Y_{52}, Y_{53}, Y_{54}$ \\
\hline
\end{tabular}
\end{center}
\end{table}
The lack of fit test is performed using the sum of squares pure error (SSPE) and the sum of squares lack of fit (SSLOF).
%$$
%SSPE = \sum_{i=1}^l\sum_{j=1}^{n_i}(Y_{ij}-\overline{Y}_{i\cdot})^2,
%$$
%$$
%SSLOF=\sum_{i=1}^l\sum_{j=1}^{n_i}(\overline{Y}_{i\cdot}-\hat{Y}_{ij})^2,
%$$
%where $Y_{i\cdot}=\sum_jY_{ij}$, $\overline{Y}_{i\cdot}=n_i^{-1}\sum_jY_{ij}$ and $\hat{Y}_{ij}$ is the fitted value.

\begin{parts}

  \PP Prove that the sum of squared errors (SSE) from fitting the
  model to the data is equal to $SSPE + SSLOF$.  For simplicity, use
  notation $Y_{i\cdot}=\sum_jY_{ij}$,
  $\overline{Y}_{i\cdot}=n_i^{-1}\sum_jY_{ij}$ and $\hat{Y}_{ij}$ as
  the fitted value.

  \PP If $SSPE=800$ and $SSLOF=6000$, perform the lack of fit test at
  significance level $\alpha=0.05$.  State the hypotheses, the test
  statistic and its numeric value, distribution of the test statistic,
  degrees of freedom of the test statistic, and the conclusion.

\end{parts}

% Steve.  applied.

\QQ Consider a $2^{6-3}$ fractional factorial design in Factors $A$,
$B$, $C$, $D$, $E$, and $F$.  An alternate fraction is used (not the
principal fraction) with the extra factor assignments as $D = -AB$, $E =
AC$, $F = -BC$.

\begin{parts}
  

\PP Specify the defining relation.



\PP Identify all main effect and two-factor
interaction aliases.  Show all work.



\PP Suppose you desired to run a second design.  This second design is
fold-over of the above design produced switching all of its
signs. What would be the generators of this second design?


\PP Suppose you create a third design which is produced by placing the
first and second designs above together (you may ignore any kind of
blocking).  What would be the defining relation of this third design?
Also comment on the resolution of this third design.

\end{parts}

%%%%%%%%%%%%%%%%%%%% Probability %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Jinfeng, STA 4321.

\QQ In any given year a male automobile policyholder will make a claim
with probability 0.3, and a female policyholder will make a claim with
probability 0.2. The fraction of the policyholders that are male is
0.6.  A policyholder is randomly chosen.

\begin{parts}
  
\PP What is the probability that the policyholder will make a claim in year 1?

\PP If the policyholder made a claim in the year 1, what is the
probability that the policyholder is a male?

\PP What is the probability that the policyholder will make a claim in
year 2, given that he/she makes a claim in year 1?

\end{parts}


% Flori. probability.

\QQ Let $X$ be a non-negative, continuous random variable.

\begin{parts}

\PP Show that 
\[ E (X) = \int_{0}^{\infty}P (X > t)dt.\]

{\it  Hint:} Show that $\int_{0}^{\infty}P (X > t)dt = \int_{0}^{\infty}xf(x)dx$, by changing order of integration after observing that  any positive $x$ can be written as $x = \int_{0}^{x}dt$.  \\

\PP Use the result stated above in (a) to show that for any
non-negative and continuous random variable $Y$ we have
\[ E(Y^n) = \int_{0}^{\infty}ny^{n-1}P(Y > y)dy.\]

{\it Hint:} Apply the result in (a) to $X = Y^n$ and make the change
of variables $t = y^n$.
  
\end{parts}

%Vic intro math stat 4321
\QQ If $Y_n = \min_{1 \leq i \leq n} X_i,$ where $X_1, X_2, \dots $
are i.i.d.\ r.v.'s with density $f(x) = \beta x^{\beta-1}$, $0 < x <
1$:

\begin{parts}
\PP What is the distribution of $Y_n? $\\
\PP Find a value of $\beta$ so that $\sqrt{n}\, Y_n$ converges in
distribution. What is the limiting distribution?  
\end{parts}


%%%%%%%%%%%%%%%%%%% Theoretical Statistics %%%%%%%%%%%%%%%%%%%%

\QQ Let $X_1, \ldots, X_n$ be a random sample from the density
\begin{displaymath}
  f(x \,|\, \mu,\sigma) = \frac{1}{\sigma}e^{(x-\mu)/\sigma}\,,\;\;\;
-\infty<x<\mu,\,\; \sigma>0\,.
\end{displaymath}
Find a two-dimensional sufficient statistic for $(\mu,\sigma)$.

% marten

\QQ Let $X_1,\ldots, X_n$ be i.i.d.\ from the density
\[ f(x|\theta)= \frac{x}{\theta^2} \exp\left(- \frac{ x^2}{ 2\theta^2} \right),\ x\ge0\]
with parameter $\theta>0$.
\begin{parts}
\PP Find the method of moments estimator of $\theta$.\\
\PP Find the maximum likelihood estimator (mle) of $\theta$.\\ 
\PP Find the asymptotic variance of the mle.

\end{parts}


% Eric
\QQ Let $X_1, X_2, \ldots, X_n$ be a random sample of gamma ($\alpha$, 1)
random variables:
$$
f(x_i)=\frac{1}{\Gamma(\alpha)}x^{\alpha-1}_ie^{-x_i}, \hspace{.1in}
\alpha>0, \hspace{.1in} x_i>0, \hspace{.1in} i=1, 2, \ldots, n.
$$
Find the maximum likelihood estimate for
$\Gamma^\prime(\alpha)/\Gamma(\alpha)$.  (Recall that $ \lim_{\alpha
  \rightarrow 0} \Gamma(\alpha)=\lim_{\alpha \rightarrow \infty}
\Gamma(\alpha)=\infty.  $)


\end{document}

